# Tag Info

### When is convergence in measure useful?

So, as a statistician, the typical use cases I've seen are, in order of "strength" as convergence and convergence in probability to a constant. Intuitively, I consider these to be the ...

### This expected value has a minimum!

Another way to do this arises from stochastic dominance. Given a positive random variable $X$ whose p.d.f is bounded above by $1$, create a uniform random variable $U[0,1]$ on the same probability ...

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1 vote

### How do we know the dual pairing between Lp spaces is well defined?

As long as $q$ is supposed to be the conjugate of $p$ (i.e. $\frac1p + \frac1q = 1$) and $p \geq 1$ then the integral of the product is well-defined thanks to Hölder's inequality, which holds for all ...
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1 vote

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### Show $0<\epsilon<1$ there exists some $\delta_{p,\epsilon}>0$ such that $m(\{x\in X:|f(x)|>\epsilon\})\geq\delta_{p,\epsilon}$ for each $f\in E_p$.

Notice that for $f\in E_p$ and $0<\varepsilon<1$, \begin{align}1&=\int_X|f|=\int_{\{|f|\leq\varepsilon\}}|f|+\int_{\{|f|>\varepsilon\}}|f|\\ &\leq \varepsilon +\|f\|_p\big(\mu(|f|>\...
• 40.7k
1 vote

### Minkowski inequality of infinite sum

Not as nice as the other answers as it requires sigma-finiteness, but here is a possibility which explores some beautiful results in measure theory! If one has that $X$ is a $\sigma$-finite space one ...
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